Combinatorics, Case Analysis / Checking Cases, Processes / Procedures

This category covers problems involving sequences of operations or steps that evolve over time or iterations. Questions might ask about the outcome of a process, whether it terminates, or properties of its state after a certain number of steps. Often related to algorithms or invariants.

• Question

  The number `458` is written on the board. In each single step, you are allowed to either multiply the number written on the board by `2`, or erase its last digit.

  Is it possible to obtain the number `14` using these operations?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  How many two-digit numbers are there such that the tens digit is greater than the units digit?

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Ali Baba and the Forty Thieves

  Ali Baba wrote the number `17` on a piece of paper. The forty thieves pass the paper to each other, and each one either adds `1` to the existing number, or subtracts `1`, until each of them has done so once, and then they return the paper to Ali Baba.

  Is it possible that the number now written on the paper is `40`?

  Topics:

  Arithmetic Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  Find all pairs of prime numbers whose difference is `17`.

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  Topics:

  Number Theory -> Prime Numbers Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  In a magical country, there are coins worth `1`, `2`, `3` and `5` liras. Yossi has `25` coins from the magical country.

  Must there necessarily be `7` coins of the same value among these coins?

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Three Friends

  There are three friends, Weiss, Schwartzman, and Rotenberg. This trio of friends is special because one of them is blonde, one is ginger (redhead), and one has black hair. One day, the blonde guy said to Schwartzman: "We are a very special trio of friends! Notice that it's not only that the last names of the three of us mean colors, but also that each of our last names does not match his hair color."

  What hair color does each of these three people have?

  Note: "Schwartzman" means "black man", "Rotenberg" means "red mountain" and "Weiss" means "white" (from Yiddish).

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables Logic -> Truth-tellers and Liars Problems
  Sources:
  • problems.ru web site

• Corruption in Parliament

  The parliament of the magical country consists of `20` people. It is known that among the `20` members of parliament, there is at least one who is not corrupt. Additionally, it is known that for any two members of parliament we choose, one is necessarily corrupt. How many members of parliament of the magical country are corrupt?

  Topics:

  Algebra -> Inequalities Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  Find a two-digit number that is twice as large as the product of its digits.

  Topics:

  Number Theory Arithmetic Algebra -> Algebraic Techniques Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations
  Sources:
  • problems.ru web site

• Question

  Solve the equation:

  `(x + 2010)(x + 2011)(x + 2012) = (x + 2011)(x + 2012)(x + 2013) `

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  K friends simultaneously learn K pieces of news (one piece of news per friend). They begin to phone each other and exchange news. Each call lasts one hour. How long will it take for all friends to know all the news? Consider the cases:
  (a) (5 points) K=64
  (b) (10 points) K=55
  (c) (12 points) K=100
  (a) Answer

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 4